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Lectu re Notes in Economics and Mathematical Systems Managing Editors: M. Beckmann and H. P. Künzi 184 Rai ner E. Bu rkard Ulrich Derigs In cooperation with T. Bönniger G. Katzakidis Assignment and Matching Problems: Solution Methods with FORTRAN-Programs Springer-Verlag Berlin Heidelberg GmbH Editorial Board H. Albach A. V. Balakrishnan M. Beckmann (Managing Editor) P. Dhrymes J. Green W. Hildenbrand W. Krelle H. P. Künzi (Managing Editor) K. Ritter R. Sato H. Schelbert P. Schönfeld Managing Editors Prof. Dr. M. Beckmann Prol. Dr. H. P. Künzi Brown University Universität Zürich Providence, RI 02912/USA CH-8090 Zürich/Schweiz Authors Rainer E. Burkard Mathematisches Institut Universität zu Köln Weyertal86 5000 Köln 41 Federal Republic of Germany Ulrich Derigs Industrieseminar Universität zu Köln Albertus-Magnus-Platz 5000 Köln 41 Federal Republic 01 Germany AMS Subject Classilications (1980): 65K05, 90BlO, 90C08, 90C09,90C35 ISBN 978-3-540-10267-0 ISBN 978-3-642-51576-7 (eBook) DOI 10.1007/978-3-642-51576-7 Thl$ work 15 subject 10 copynght. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use 01 illustrations, broadcasting, reproduction by photocopying machine orßimilar mearos. and storage in data banks. Under § 54 01 the German COPYright Law where copies are made for other than private use, a fee is payable 10 the publisher, the amount 01 the fee 10 be determmed by agreement with the publisher. 9 by Sprirger-Verlaq Berhn Heidelberg 1980 Originally published by Springer-Verlag Berlin Heidelberg in 1980. 2141/3140-543210 Assignment and matching problems belang to those cornbinatorial opti mization problems which are weIl understood in theory and have many applications in practice. Since a research group qf the Mathematical Institute in Cologne has worked in this field for several years, we feIt that a publication of the developed procedures and algorithms would be useful, both for further research and academic tu': tion as weIl as for applications in practice. This book covers linear assignment problems wi th SUfi and bottleneck objective functions, cardinality matching problems, perfect matching problems with SUffi and bottleneck objective functions, the Chinese postman problem and optimal as weIl as heuristic algorithms for the quadratic assignment problem. Every problem will be first described in short and then a FORTRAN-routine is given for it, which is docu mented in detail and illustrated by test exarnples. Many helped us through the years to develop efficient codes for the above rnentioned problems. Our special thanks go to T. Bönniger and G. Katzakidis who helped us in compiling the final vers ions and by performing extensive computational tests of the computer programs. The development of these computer codes would not have been possible without the excellent facilities of the Computer Center of the Uni versity of Cologne. All programs were thoroughly tested on the CDC CYBER 76 installation of this computer center. Independent test runs were perfarmed on the IBM 4331 installation in the Institut for Ökonometrie and Operations Research, University of Bonn. We are very indepted to Prof. B. Korte and Dr. Grätschei, Bann, for their assist ance in having these test runs perforrned. Further we thank Prof. Th. Meis, Köln, for the permission to use his sorting routine SSORT as a subroutine in our programs . Various drafts of this manuscript were type.d by Miss B. Hünten and Mrs. E. Rath who have hecome cryptologists by deciphering our hand writings. Many thanks to thern! The authors are grateful for any comment and report on the usage of the \iescribed programs . May they turn out to be useful! COlogne, August 1980 R.E.B. and U.D. TABLE OF CONTRNTS Page Introduction 1. The Linear Sum Assignment Problem 2. The Linear Bottleneck Assignment Problem 16 3. The Cardinality Matching Problem 25 4. The Sum Matehing Problem 37 5. The Bottleneck Matehing Problem 60 6. The Chinese Postman Problem 72 7. Quadratic Assignrnent Problems 99 8. QAP Heuristic 1: The method of increasing 120 degree of freedorn 9. QAP Heuristic 2: Cutting plane and exchange 127 method 10. General Subroutines 146 In troduction In this book we present a number of optimization technigues and compu ter codes which have been designed for assignment and rnatching problems. We consider as weIl the classical SUffi objective as the bottleneck objec tive and for the matching problem the pure cardinality case, tao. Every problem resp. code is described in a special chapter which is self-contained.Thus the reader will be able tci apply a special routine without reading the whole book. All chapters are identically structured following the scheme (a) Formulation of the problem (bi The algorithm (c) References (d) Description of the program (e) Program listings and sample output In (a) we introduce the specific problem and state same alternative for mulations thereof.ln (b) the basic ideas of the implemented algorithm are roughly described.The mathematical background is only sketched.The reader can find detai ls in the references, lis ted in section (c) of ev ery chapter.ln (d) we give directions for using the code.The reader will find he re informations on the structure of the program and a de scription of the INPUT and OUTPUT parameters with the respective dimen sion requirements .The total storage requirement and the running time for medium sized problems are given.The running time was determined either by testing a great variety of randomly genera ted problems or by standard problems from the literature on a CDC CYBER 76. In (e) the reader finds the complete list of the FORTRAN IV subroutines . These subroutines start .wi th a cornment where all parameters, arrays and variables are defined. A MAIN-program is added to describe the use of the suhroutine and to enable the solution of the given sample problem. All computer codes in this book are wri tten in ANSI FORTR1\N and they are therefore machine independent.The codes were developed and extensively tested on the COC CYBER 76 of the Computer Center of the University of Cologne.Further tests were run on an IBM 4331 of the Sonderforschungs bereich 21 at the University of Bann. 1. The Linear Surn Assignment Problem a) Formulation of the problem Let the finite set N {1,2, ... ,n}, nE:JN, be given. A bijective (one to one) mapping tp: N -+ N is called apermutation of the set N. The set of all permutations <p of the set N is denoted by f n' Let an integer/real (nxn) matrix C = (cij) of cast elements cij be given. Then we can associate with every permutation tp E: fn the costs The Linear Sum Assignment Problem (LSAP) is to find apermutation tp E: fn with minimal costs,i.e. (1. 1) A graph theoretic interpretation of LSAP can be given as follows. For <pE!n the graph M<p is a partial graph of the complete bipartite graph Kn,n such that every node is incident to exactly one edge cf Mtp' Such graphs are called perfeet matchings. So LSAP is equivalent to the problem of finding a minimal weighted perfeet matching in an edge-weighted bipartite graph. (An algorithm for solving perfect matching problems in general (not necessarily bipartite) graphs is presented in section 4) • A cornmonly used formulation of L5AP is the following min r r c .. x .. subject to iEN JEN ~J ~J (1.2) r x .. 1, for iEN JEN lJ (1.3) l: x .. 1, for JEN iEN ~J (1.4) xij .:: 0, for i E N, j E N. The polyhedron PA defined by (1.2) - (1.4) is called the assignment poZytope. It is weIl known that PA has only vertices with (0,1) compo nents 'tlhich in turn correspond to the permutations cf fn· b) The algorithm The fOllowing algorithms for solving LSAP was proposed by TOMIZAWA [3] and improved by DORHOUT [2]. During the algorithm so calledk-problems (Pk) are solved for k == 1,2, ... ,n. Let f'n(K) denote the set of all one to one mappings 4>k: K ~ N for K = (1 ,2, ••• ,k) and kEN. Then the k-problem is defined by: (Pk) min L ci,w (i) 4>Efn (K) iEK The solution of (P1) is obvious. Starting with a solution of (Pk) for 1 2 k~n-l the solution ~ of (Pk+1) is constructed by means of a shortest augmenting path. Here an augmenting path is a sequence ß~ of mutually distinct matrix entries with j1 = 4>(i1) for 1 == 1, ... , r jr+1 t <j>(K). Then we find \)J(k+1)= j1 for 1 = 1,2, ... Ir for iEK'{ i1 I 1=1, ... ,r} . Figure 1 shows such an augmenting path. Here the circled matrix entries represent the mapping Wk E f n (K) . e k+1 We define c(4) 44» := L C i=l i,~di) . • Lr • • • • k+l Figure 1: Augmenting path Let Dk be the set of all augmenting paths with respect to <j) E :f n (Kl • Then .6.\jJ EDk is called shartest auqmenting path if c (tP ® tHO) 1s minimal. Such a shartest augrnenting path can be faund using a modification of the shortest path algorithm of DIJKSTRA [1]. With an optimal solu tion ~k for (Pk) and a shartest augmenting path .6.\jJ we obtain l\J = <j) G) LI<j) as optimal solution for (Pk+11. After the augmentation the cast coefficients of the matrix C =(Cij) have to be transformed. This is done by rneans of a vector y E m2n whose elements can be interpreted as dual variables. After termination of the algorithm the vector y represents the associated dual solution i.e. for i, JEN Computational experiments show that the running time of the algorithm can be decreased by means of astart heuristic in which an initial set K ~ N and a mapping l.Ok E::fn (K) are determined in the following way: Par t;!"lery i € N we calculate Si := min C •• JEN 1) Then ~e calculate t). := min (ci)·-si) for every JEN iEN Every entry (i, j) wi th cij -si -tj==O is called admissible. Inspecting the rows first and the columns thereafter we deterrnine a maximal set in K ::: N and an associated mapping <llk E (k) such that (i, <llk (i» is ad.missible for i E K. Then the shortest augmenting path rnethod is star ted with respect to <llk' c) References [1] Dijkstra, E.W.: A note on two problems in connection with graphs. Numerische Mathematik.l (1959), 269-271. [2] Dorhout, B.: Het lineaire toewij zingsproblem, vergelijking van algoritrnen. Report BN 21, Stichting Mathe matisches Centrum, Amsterdam, 1973. (3} Tomizawa, N.: On some techniques useful for solution of transportation network problems. Networks (1972), 179-194. d) Description of the program The program for solving the linear sum assignment problem consists of a MAIN-prograrn for INPUT and OUTPUT operations and a single sub routine LSAP. In MAIN all arrays and vectors are specified in DIMENSION statements for the test problem given below. They rnay be altered for the parti- cular problems. INPUT: As input data the dimension n of the problem, the cost matrix C and a parameter SUP are required. The cast coeffi eients cij should be nonnegative integers. The matrix C = (Cij) has to be read row by row, i.e. and is then stored on a vector C(I) of length n2• The parameter SUP has to be a sufficiently large number, i.e. SUP ~ 1: 1: ciJ. • iEN JEN OUTPUT: The MAIN program prints out the name of the problem and the cost matrix C = (cij). The optimal permutation <jl is stored on the vector SPALTE(I) and the optimal assignment is pr int ed out in the form <jl(i) , i = 1,2, ... ,n. Then the value 1: of the optimal permutation iEN is given. The associated dual solution is stored on the two vectors YS(I) and YT(J) of length n, but is not printed out here. The program requires a storage capacity of n2 + Sn. Computational tests with randomly generated problems on a CDC CYBER 76 showed that the expected (= mean) CPU-time is for n 100 Rf 0.1 sec and for n 200 .. 0.5 sec.